Plancherel and Fourier inversion

For $f\in\mathcal S(\mathbb R^d)$, both its transform and its inverse integral are absolutely integrable in the relevant variables, and Fourier inversion recovers $f(x)$ for every $x$. One rigorous strategy inserts a Gaussian frequency cutoff, uses Fubini on the regularized integral, identifies a spatial approximate identity, and then removes the cutoff. The approximation argument matters: naively substituting one unregularized oscillatory integral into another and declaring the inner integral a point mass is not an application of ordinary Fubini.

A useful broader theorem assumes $f\in L^1(\mathbb R^d)$ and $\widehat f\in L^1(\mathbb R^d)$. The inverse integral then defines a continuous representative equal to $f$ almost everywhere. At every point where the chosen input representative is continuous, the same value is recovered. Without the second integrability hypothesis, that inverse integral need not converge absolutely. A regularized or distributional inversion may still exist, but it is a different assertion requiring its own interpretation.

On Schwartz functions, the inner product is preserved by the Fourier transform. In particular the spatial and frequency squared norms agree, without an additional prefactor under this normalization. Since Schwartz functions are dense in L², approximating any L² input by them gives a Cauchy sequence of transforms with the same distances. Its L² limit defines the extended transform independently of the approximating sequence. Extending the inverse in the same way makes the transform unitary and onto the frequency L² space.

An L² function need not be integrable, so its transform cannot always be defined by the original absolute integral. Even when the input is also integrable and its ordinary transform exists, that transform need not be integrable enough for absolute inverse integration. Norm inversion nevertheless recovers the input equivalence class. Pointwise assertions require further conditions or a specified limiting procedure. Unitarity is a powerful statement about average energy and stability, but no point evaluation is a bounded operation on the entire L² space.

Once an explicit transform has been found for an integrable square-integrable input, Plancherel evaluates its frequency energy without directly integrating an oscillatory squared expression. This is especially useful for interval indicators, whose transforms decay like a reciprocal frequency and need not have absolutely integrable magnitude. The squared magnitude is integrable, exactly as the theorem predicts. The removable value at zero frequency is the input's total mass, and getting it right checks both the scale and normalization.

Energy preservation also controls error: replacing a signal by an L² approximation produces exactly the same L² error between their transforms. It does not preserve every norm or every feature. A small-energy spike can have large pointwise amplitude, and its transform can spread over a broad frequency range. When an application needs a peak, a derivative, or a tail integral, ask for bounds in the relevant space rather than treating the L² identity as a universal guarantee of reconstruction quality.